<description/> <author>Константин Титов</author> <company/> <keywords/> <revisedBy>Константин Титов</revisedBy> </userData> <identityInfo> <revision>5</revision> <documentID>83c8a47e-0ce9-4b59-ab89-f75d3acffec3</documentID> <versionID>fe9b24fa-9279-4056-8f23-65e53ca7dfbb</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="14" font-weight="bold" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 2"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="12" font-weight="bold" font-style="italic" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 3"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="12" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Paragraph"> <blockAttr margin-left="0" margin-right="0" text-indent="21" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="List"> <blockAttr margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Indent"> <blockAttr margin-left="108" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Title"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="24" font-weight="bold" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Subtitle" base-style="Title"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="18" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="narrow-dot" derivative="derivative" literal-subscript="large" definition="colon-equal" global-definition="triple-equal" local-definition="left-arrow" equality="bold-equal" symbolic-evaluation="right-arrow"/> <mathStyles> <mathStyle name="Variables" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Constants" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 1" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 2" font-family="Courier New" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 3" font-family="System" font-charset="204" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 4" font-family="Script" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 5" font-family="Roman" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 6" font-family="Modern" font-charset="255" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 7" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="charge" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" imaginary-value="i" zero-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="56.63622" right="14.14488" top="70.86614" bottom="70.86614"/> <header use-full-page-width="true"> <left>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fswiss\fprq2\fcharset204{\*\fname Arial;}Arial CYR;}{\f1\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs16\{p\}\par \{f\}\cf0\f1\fs18\par } </left> <center>{\rtf1\ansi\ansicpg1251\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq14 Arial;}} \viewkind4\uc1\pard\qc\f0\fs18\{n\}\par } </center> </header> <footer use-full-page-width="true"> <left>{\rtf1\ansi\ansicpg1251\deff0\deflang1033{\fonttbl{\f0\froman\fcharset204{\*\fname Times New Roman;}Times New Roman CYR;}{\f1\froman Times New Roman;}} \viewkind4\uc1\pard\lang1049\f0\fs24\'d2\'e8\'f2\'ee\'e2 \'ca.\'c2.\lang1033\f1\par } </left> <center>{\rtf1\ansi\ansicpg1251\deff0\deflang1033{\fonttbl{\f0\froman Times New Roman;}} \viewkind4\uc1\pard\qc\f0\fs24\{n\}\par } </center> <right>{\rtf1\ansi\ansicpg1251\deff0\deflang1033{\fonttbl{\f0\froman Times New Roman;}} \viewkind4\uc1\pard\qr\f0\fs24\{d\}\par } </right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="200" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="187" left="12" top="12.75" width="468" height="27" align-x="12" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Приведем еще один из вариантов алгоритма неявной схемы метода сеток в среде Mathcad </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">(метод прогонки)</inlineAttr> </f> </text> </region> <region region-id="183" left="18" top="54" width="120.75" height="27.75" align-x="18" align-y="66" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Введем обозначения:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp count="2"/>- константы:</inlineAttr> </f> </text> </region> <region region-id="178" left="186" top="68.25" width="20.25" height="13.5" align-x="194.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">l</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="179" left="222" top="68.25" width="23.25" height="13.5" align-x="233.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="180" left="258" top="66.75" width="66.75" height="13.5" align-x="258" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">;<sp count="3"/>дискреты:</inlineAttr> </f> </text> </region> <region region-id="181" left="342" top="68.25" width="30" height="13.5" align-x="353.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="182" left="384" top="61.5" width="35.25" height="27.75" align-x="395.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>600</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="24" left="12" top="90.75" width="500.25" height="27" align-x="12" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">переменные: х - расстояние, t - время; их дискретные значения: x=h*i,<sp count="2"/>t=s*j, где индексы i и j принимают значения целых положительных чисел (см. ниже).</inlineAttr> </f> </text> </region> <region region-id="26" left="12" top="126.75" width="105.75" height="13.5" align-x="12" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Другие параметры:</inlineAttr> </f> </text> </region> <region region-id="28" left="132" top="121.5" width="35.25" height="27.75" align-x="143.25" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">d</ml:id> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>600</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="29" left="186" top="115.5" width="42" height="39.75" align-x="198.75" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ω</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="117" left="258" top="128.25" width="30" height="13.5" align-x="270" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ω</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000004</ml:real> </result> </ml:eval> </math> <rendering item-idref="8"/> </region> <region region-id="175" left="342" top="123.75" width="52.5" height="18" align-x="366" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">TOL</ml:id> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-5</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="41" left="12" top="175.5" width="438" height="53.25" align-x="12" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Уравнение теплопроводности в частных производных: </inlineAttr> </f> <region region-id="40" left="12" top="189.75" width="137.25" height="39" align-x="54.75" align-y="213" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="10"/> </region> </text> </region> <region region-id="43" left="12" top="211.5" width="499.5" height="54" align-x="12" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">При замене частных производных разностными отношениями уравнение теплопроводности можно записать в виде:<sp count="3"/> </inlineAttr> </f> <region region-id="44" left="262.5" top="225.75" width="216.75" height="39.75" align-x="282" align-y="248.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">ω</ml:id> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">ω</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="11"/> </region> </text> </region> <region region-id="55" left="12" top="277.5" width="344.25" height="27.75" align-x="12" align-y="288" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Зададим краевые условия:<sp count="3"/> </inlineAttr> </f> <region region-id="54" left="12" top="291.75" width="187.5" height="13.5" align-x="138.75" align-y="301.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:range> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:range> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:range> </math> <rendering item-idref="12"/> </region> </text> </region> <region region-id="58" left="12" top="306.75" width="21" height="13.5" align-x="12" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">где:</inlineAttr> </f> </text> </region> <region region-id="62" left="42" top="303" width="71.25" height="18.75" align-x="63.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="63" left="150" top="308.25" width="45" height="13.5" align-x="171.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">φ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:real>60</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="64" left="222" top="308.25" width="36.75" height="13.5" align-x="246.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">ψ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="69" left="282" top="308.25" width="98.25" height="13.5" align-x="315.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">g</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>36</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="73" left="12" top="331.5" width="470.25" height="41.25" align-x="12" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Перейдем к дискретным переменным. Зная диапазоны их изменения: </inlineAttr> </f> <region region-id="71" left="12" top="345.75" width="81" height="13.5" align-x="51" align-y="355.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">l</ml:id> </ml:apply> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">d</ml:id> </ml:apply> </ml:range> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">можно определить </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">диапазоны изменения</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> индексов i,j как:</inlineAttr> </f> </text> </region> <region region-id="79" left="18" top="372.75" width="60.75" height="29.25" align-x="32.25" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="82" left="96" top="372.75" width="57.75" height="29.25" align-x="107.25" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">d</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="83" left="180" top="380.25" width="38.25" height="13.5" align-x="188.25" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="90" left="240" top="380.25" width="36" height="13.5" align-x="249" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="92" left="312" top="378.75" width="34.5" height="13.5" align-x="312" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Тогда:</inlineAttr> </f> </text> </region> <region region-id="93" left="360" top="380.25" width="36" height="13.5" align-x="373.5" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="94" left="426" top="380.25" width="28.5" height="13.5" align-x="436.5" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="95" left="18" top="414.75" width="491.25" height="81" align-x="18" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Решение уравнения теплопроводности в разностном виде запишем в виде таблицы, где первая строка и крайние столбцы представляют собой соответственно краевые условия:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">распределение температуры u(x,0) в начальный момент времени (первая строка), ее значения на концах стержня u(0,t), u(l,t) (крайние столбцы). </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Начнем с краевых условий (здесь первый индекс обозначает номер строки таблицы, второй - </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">номер</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> столбца) :</inlineAttr> </f> </text> </region> <region region-id="97" left="18" top="518.25" width="53.25" height="15.75" align-x="42" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="99" left="102" top="518.25" width="57" height="15.75" align-x="126.75" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="100" left="192" top="518.25" width="63.75" height="15.75" align-x="220.5" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="102" left="282" top="505.5" width="87.75" height="39.75" align-x="303" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="189" left="24" top="565.5" width="459" height="46.5" align-x="36" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="2" cols="11"> <ml:real>0</ml:real> <ml:real>0.3</ml:real> <ml:real>0.099</ml:real> <ml:real>0</ml:real> <ml:real>0.192</ml:real> <ml:real>0</ml:real> <ml:real>0.273</ml:real> <ml:real>0</ml:real> <ml:real>0.336</ml:real> <ml:real>0</ml:real> <ml:real>0.375</ml:real> <ml:real>0</ml:real> <ml:real>0.384</ml:real> <ml:real>0</ml:real> <ml:real>0.357</ml:real> <ml:real>0</ml:real> <ml:real>0.28799999999999992</ml:real> <ml:real>0</ml:real> <ml:real>0.17099999999999996</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <general precision="5" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="4"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false"/> <table item-idref="28"/> </resultFormat> </math> <rendering item-idref="29"/> </region> <region region-id="110" left="18" top="654.75" width="476.25" height="27" align-x="18" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Продолжаем дальше вычисления и записываем их в таблицу. Для этого сформулируем алгоритм метода прогонки:</inlineAttr> </f> </text> </region> <region region-id="192" left="24" top="702.75" width="435" height="13.5" align-x="24" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Рассчитываемые коэффициенты при прямом ходе:</inlineAttr> </f> </text> </region> <region region-id="196" left="36" top="740.25" width="54" height="13.5" align-x="44.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>2</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="197" left="126" top="733.5" width="51" height="28.5" align-x="143.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">ω</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="198" left="222" top="730.5" width="75" height="33.75" align-x="237" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="199" left="36" top="774.75" width="113.25" height="13.5" align-x="36" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Алгоритм имеет вид:</f> </text> </region> <region region-id="164" left="24" top="819" width="243.75" height="213" align-x="36.75" align-y="834" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R</ml:id> <ml:program> <ml:localDefine> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">W</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">U</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">U</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">q</ml:id> <ml:range> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">q</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">q</ml:id> </ml:sequence> </ml:apply> </ml:localDefine> </ml:for> <ml:for> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:real>0</ml:real> </ml:sequence> 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из вариантов алгоритма неявной схемы метода сеток в среде Mathcad </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">(метод прогонки)</inlineAttr> </f> </text> </region> <region region-id="183" left="18" top="54" width="120.75" height="27.75" align-x="18" align-y="66" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Введем обозначения:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp count="2"/>- константы:</inlineAttr> </f> </text> </region> <region region-id="178" left="186" top="68.25" width="20.25" height="13.5" align-x="194.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">l</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="179" left="222" top="68.25" width="23.25" height="13.5" align-x="233.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">a</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="180" left="258" top="66.75" width="66.75" height="13.5" align-x="258" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">;<sp count="3"/>дискреты:</inlineAttr> </f> </text> </region> <region region-id="181" left="342" top="68.25" width="30" height="13.5" align-x="353.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="182" left="384" top="61.5" width="35.25" height="27.75" align-x="395.25" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>600</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="24" left="12" top="90.75" width="500.25" height="27" align-x="12" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">переменные: х - расстояние, t - время; их дискретные значения: x=h*i,<sp count="2"/>t=s*j, где индексы i и j принимают значения целых положительных чисел (см. ниже).</inlineAttr> </f> </text> </region> <region region-id="26" left="12" top="126.75" width="105.75" height="13.5" align-x="12" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Другие параметры:</inlineAttr> </f> </text> </region> <region region-id="28" left="132" top="121.5" width="35.25" height="27.75" align-x="143.25" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">d</ml:id> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>600</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="29" left="186" top="115.5" width="42" height="39.75" align-x="198.75" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ω</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="117" left="258" top="128.25" width="30" height="13.5" align-x="270" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ω</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.0000000000000004</ml:real> </result> </ml:eval> </math> <rendering item-idref="8"/> </region> <region region-id="175" left="342" top="123.75" width="52.5" height="18" align-x="366" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">TOL</ml:id> <ml:apply> <ml:pow/> <ml:real>10</ml:real> <ml:real>-5</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="41" left="12" top="175.5" width="438" height="53.25" align-x="12" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Уравнение теплопроводности в частных производных: </inlineAttr> </f> <region region-id="40" left="12" top="189.75" width="137.25" height="39" align-x="54.75" align-y="213" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="10"/> </region> </text> </region> <region region-id="43" left="12" top="211.5" width="499.5" height="54" align-x="12" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">При замене частных производных разностными отношениями уравнение теплопроводности можно записать в виде:<sp count="3"/> </inlineAttr> </f> <region region-id="44" left="262.5" top="225.75" width="216.75" height="39.75" align-x="282" align-y="248.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">ω</ml:id> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">ω</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </math> <rendering item-idref="11"/> </region> </text> </region> <region region-id="55" left="12" top="277.5" width="344.25" height="27.75" align-x="12" align-y="288" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Зададим краевые условия:<sp count="3"/> </inlineAttr> </f> <region region-id="54" left="12" top="291.75" width="187.5" height="13.5" align-x="138.75" align-y="301.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:range> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:range> <ml:apply> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">u</ml:id> <ml:sequence> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:range> </math> <rendering item-idref="12"/> </region> </text> </region> <region region-id="58" left="12" top="306.75" width="21" height="13.5" align-x="12" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">где:</inlineAttr> </f> </text> </region> <region region-id="62" left="42" top="303" width="71.25" height="18.75" align-x="63.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">x</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="63" left="150" top="308.25" width="45" height="13.5" align-x="171.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">φ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:real>60</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="64" left="222" top="308.25" width="36.75" height="13.5" align-x="246.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">ψ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="69" left="282" top="308.25" width="98.25" height="13.5" align-x="315.75" align-y="318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">g</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>36</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="73" left="12" top="331.5" width="470.25" height="41.25" align-x="12" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Перейдем к дискретным переменным. Зная диапазоны их изменения: </inlineAttr> </f> <region region-id="71" left="12" top="345.75" width="81" height="13.5" align-x="51" align-y="355.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:range xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">l</ml:id> </ml:apply> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:id xml:space="preserve">d</ml:id> </ml:apply> </ml:range> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">можно определить </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">диапазоны изменения</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> индексов i,j как:</inlineAttr> </f> </text> </region> <region region-id="79" left="18" top="372.75" width="60.75" height="29.25" align-x="32.25" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">l</ml:id> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="82" left="96" top="372.75" width="57.75" height="29.25" align-x="107.25" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">d</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="83" left="180" top="380.25" width="38.25" height="13.5" align-x="188.25" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">m</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="90" left="240" top="380.25" width="36" height="13.5" align-x="249" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="92" left="312" top="378.75" width="34.5" height="13.5" align-x="312" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Тогда:</inlineAttr> </f> </text> </region> <region region-id="93" left="360" top="380.25" width="36" height="13.5" align-x="373.5" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>10</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="94" left="426" top="380.25" width="28.5" height="13.5" align-x="436.5" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="23"/> </region> <region region-id="95" left="18" top="414.75" width="491.25" height="81" align-x="18" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Решение уравнения теплопроводности в разностном виде запишем в виде таблицы, где первая строка и крайние столбцы представляют собой соответственно краевые условия:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">распределение температуры u(x,0) в начальный момент времени (первая строка), ее значения на концах стержня u(0,t), u(l,t) (крайние столбцы). </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Начнем с краевых условий (здесь первый индекс обозначает номер строки таблицы, второй - </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">номер</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> столбца) :</inlineAttr> </f> </text> </region> <region region-id="97" left="18" top="518.25" width="53.25" height="15.75" align-x="42" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="99" left="102" top="518.25" width="57" height="15.75" align-x="126.75" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">φ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="100" left="192" top="518.25" width="63.75" height="15.75" align-x="220.5" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ψ</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="102" left="282" top="505.5" width="87.75" height="39.75" align-x="303" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="189" left="24" top="565.5" width="459" height="46.5" align-x="36" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">U</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="2" cols="11"> <ml:real>0</ml:real> <ml:real>0.3</ml:real> <ml:real>0.099</ml:real> <ml:real>0</ml:real> <ml:real>0.192</ml:real> <ml:real>0</ml:real> <ml:real>0.273</ml:real> <ml:real>0</ml:real> <ml:real>0.336</ml:real> <ml:real>0</ml:real> <ml:real>0.375</ml:real> <ml:real>0</ml:real> <ml:real>0.384</ml:real> <ml:real>0</ml:real> <ml:real>0.357</ml:real> <ml:real>0</ml:real> <ml:real>0.28799999999999992</ml:real> <ml:real>0</ml:real> <ml:real>0.17099999999999996</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <general precision="5" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="4"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false"/> <table item-idref="28"/> </resultFormat> </math> <rendering item-idref="29"/> </region> <region region-id="110" left="18" top="654.75" width="476.25" height="27" align-x="18" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Продолжаем дальше вычисления и записываем их в таблицу. Для этого сформулируем алгоритм метода прогонки:</inlineAttr> </f> </text> </region> <region region-id="192" left="24" top="702.75" width="435" height="13.5" align-x="24" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Рассчитываемые коэффициенты при прямом ходе:</inlineAttr> </f> </text> </region> <region region-id="196" left="36" top="740.25" width="54" height="13.5" align-x="44.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>2</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="197" left="126" top="733.5" width="51" height="28.5" align-x="143.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">ω</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="198" left="222" top="730.5" width="75" height="33.75" align-x="237" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="199" left="36" top="774.75" width="113.25" height="13.5" align-x="36" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Алгоритм имеет вид:</f> </text> </region> <region region-id="164" left="24" top="819" width="243.75" height="213" align-x="36.75" align-y="834" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R</ml:id> <ml:program> <ml:localDefine> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">W</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">U</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">U</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">q</ml:id> <ml:range> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">q</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:id xml:space="preserve">q</ml:id> </ml:sequence> </ml:apply> </ml:localDefine> </ml:for> <ml:for> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">ω</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>2</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">i</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">ω</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">j</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">g</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> <ml:for> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:id xml:space="preserve">j</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:localDefine> </ml:for> </ml:program> </ml:for> <ml:id xml:space="preserve">W</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="166" left="312" top="834.75" width="96" height="13.5" align-x="312" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Краевые условия</inlineAttr> </f> </text> </region> <region region-id="168" left="318" top="924.75" width="179.25" height="29.25" align-x="318" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Формирование вектора коэффи-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">циентов </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="14"> <inlineAttr line-through="false">В </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">при прямом ходе</inlineAttr> </f> </text> </region> <region region-id="200" left="318" top="978.75" width="191.25" height="40.5" align-x="318" align-y="990" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Расчет вектора результатов, запи-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">сываемых в строку "j" таблицы при</inlineAttr> </f> <f family="Arial Cyr" 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